Problem 12 - College Algebra

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(3 x^{3}+4 x^{2}-8 x+2\right) \div(x-3) $$

Step-by-Step Solution

Answer

The remainder is 95.

Step 1: Identify the polynomial and divisor

The given polynomial is \( 3x^3 + 4x^2 - 8x + 2 \) and the divisor is \( x - 3 \). According to the Remainder Theorem, the remainder of a polynomial \( f(x) \) when divided by \( x-a \) is \( f(a) \). In this case, \( a = 3 \).

Step 2: Substitute the root into the polynomial

To find the remainder, substitute \( x = 3 \) into the polynomial \( f(x) = 3x^3 + 4x^2 - 8x + 2 \). Calculate \( f(3) \).

Step 3: Calculate each term

1. Compute \( 3(3)^3 = 3 \times 27 = 81 \).2. Compute \( 4(3)^2 = 4 \times 9 = 36 \).3. Compute \( -8(3) = -24 \).4. The constant term is \( +2 \).

Step 4: Sum the results

Add the values obtained from each term: \( 81 + 36 - 24 + 2 = 95 \).

Step 5: Conclusion

The remainder when \( 3x^3 + 4x^2 - 8x + 2 \) is divided by \( x-3 \) is \( 95 \), according to the Remainder Theorem.

Key Concepts

Polynomial Division

Polynomial division is similar to numerical division, where we divide one polynomial by another. In this case, we are concerned with dividing a cubic polynomial by a linear polynomial. The process involves dividing the terms of the polynomial based on their degree.

• The highest degree term is divided by the highest degree term of the divisor.
• Subtract the result from the original polynomial.
• Repeat the process for the new polynomial formed after subtraction, using the next highest degree term.

This process continues until the degree of the new polynomial is less than the degree of the divisor. The final polynomial, which can't be divided further, is considered the remainder. In numerical terms, it may resemble the old-school long division method. However, finding the remainder through division can sometimes be cumbersome, which is why we often prefer other shortcuts like the Remainder Theorem.

Remainder Calculation

Remainder calculation using the Remainder Theorem provides a shortcut to this division task by substituting the root of the divisor into the polynomial. Instead of dividing step-by-step, we can jump straight to finding the remainder.

• Identify that the divisor is in the form \( x - a \).
• Substitute \( a \) into the original polynomial \( f(x) \).
• Simplify the expression to calculate the final remainder.

Using the provided exercise as an example, the divisor is \( x - 3 \), so we substitute \( 3 \) into the cubic polynomial. This method combines insight from algebra and arithmetic, as it relates the idea of evaluating expressions with division's remainder aspect.

Substitution Method

The substitution method in the context of the Remainder Theorem is straightforward and efficient for remainder calculations. This technique involves inserting specific values in place of a variable, simplifying the equation and showing practical applications of polynomial evaluations.
For the given polynomial \( 3x^3 + 4x^2 - 8x + 2 \), you substitute \( x = 3 \) to directly calculate the remainder.

• Calculate each term separately and sum them for \( f(3) \).
• This might involve multiple arithmetic operations (e.g., power raising, multiplication).
• Finally, add or subtract constants as needed.

This approach demonstrates how substitution can simplify otherwise complex tasks. It's a powerful method, especially when dealing with polynomial expressions, enabling us to find results quickly and accurately.